The perturbation \(\Delta \)  can represent various types of uncertainties and nonlinearities, including sector bounded nonlinearities, slope restricted nonlinearities, and unmodeled dynamics. Thus considering \(\Delta \)  extends our framework to the class of plants beyond LTI plants. The input-output relationship of \(\Delta \)  is characterized with an integral quadratic constraint (IQC) [1]}, which consists of a filter \(\Psi \)  applied to the input \(p\)  and output \(q\)  of \(\Delta \) , and a constraint on the output \(r\)  of \(\Psi \) . The filter \(\Psi \)  is an LTI system with the zero initial condition \(\psi (0) = 0_{n_{\psi \times 1}}\) :
\(\begin{array}{ll}\psi (k+1) &= A_\psi \ \psi (k) + B_{\psi 1} \ p(k) + B_{\psi 2} \ q(k), \\r(k) &= C_{\psi } \ \psi (k) + D_{\psi 1} \ p(k) + D_{\psi 2} \ q(k).\end{array}\) 
